"Runs" and "Home Runs" are two different terms.
A "home run" is a "run." A "run" isn't always a "home run."
According to the 2016 MLB Official Rules, a "run" is:
the score made by an offensive player who
advances from batter to runner and touches first, second, third and home
bases in that order.
What does Home Runs Allowed mean?
A "home run" is(1):
scored when the ball is hit in such a way that the batter is able to
circle the bases and reach home safely in one play without any errors
being committed by the defensive team in the process.
Typical usage is a fair ball that flies over the wall in the outfield, but there are instances in which the batter can still score a run in one play without doing this (called an "inside-the-park home run")(2). "Home Runs Allowed" is the number of times the pitching team allows opposing batters to score a run in one play.
What is the concept of allowing it to happen?
"Home Runs Allowed" is the number of times the pitching team allows opposing batters to score a run in one play. As you state in comments, "allowed" can be interchangeable with "conceded."
Does a high value for Runs Allowed for a team mean that when it was pitching it permitted the batting team to get a lot of home runs?
Not necessarily. The batting team may have gotten a combination of singles, doubles, triples, walks, errors, etc. during an inning to score several runs. A 2011 game (box score) shows the Mets scored 14 runs in a game without a home run.
